chemia - Studia

CLUJ AND RELATED POLYNOMIALS IN TORI
(iii) Pair-wise product; the polynomial is called Cluj-Product (and
symbolized CJP) [4,8,15-19] or also Szeged (and symbolized SZ) [12-14]:
v ( v−v
)
CJP( x) = SZ( x)
k k
=∑ x (3)
e
Their coefficients can be calculated from the graph proximities /
semicubes as shown in Figure 1: the product of three numbers (in the front
of brackets – right hand of Figure 1) has the meaning: (i) symmetry of G;
(ii) occurrence of c k (in the whole structure) and (iii) e k .
The first derivative (in x=1) of a (graph) counting polynomial provides
single numbers, often called topological indices.
Observe that the first derivative (in x=1) of the first two polynomials
gives one and the same value (Figure 1), however, their second derivative
is different and the following relations hold in any graph [4,7].
CJS′ (1) = PI ′
v
(1) ; CJS′′ (1) ≠ PI ′′
v
(1)
(4)
The number of terms, given by P(1), is: CJS(1)=2e while PI v (1)=e
because, in the last case, the two endpoint contributions are pair-wise summed
for any edge in a bipartite graph.
In bipartite graphs, the first derivative (in x=1) of PI v (x) takes the
maximal value:
PI ′
v
(1) = e ⋅ v = | E( G)| ⋅| V( G )|
(5)
It can also be seen by considering the definition of the corresponding
index, as written by Ilić [20]:
PI ′
v( G) = PIv (1) = ∑ nu, v
+ nv, u
= V ⋅ E − ∑ mu,
v
e= uv
e=
uv
(6)
where n u,v , n v,u count the non-equidistant vertices with respect to the
endpoints of the edge e=(u,v) while m(u,v) is the number of equidistant
vertices vs. u and v. However, it is known that, in bipartite graphs, there are
no equidistant vertices, so that the last term in (6) will disappear. The value
of PI v (G) is thus maximal in bipartite graphs, among all graphs on the same
number of vertices; the result of (5) can be used as a criterion for the “bipatity”
of a graph [6].
The third polynomial is calculated as the pair-wise product; notice
that Cluj-Product CJP(x) is precisely the (vertex) Szeged polynomial SZ v (x),
defined by Ashrafi et al. [12-14] This comes out from the relations between
the basic Cluj (Diudea [16,21,22]) and Szeged (Gutman [22,23]) indices:
CJP′ (1) = CJDI( G) = SZ( G) = SZ ′
v
(1)
(7)
All the three polynomials (and their derived indices) do not count
the equidistant vertices, an idea introduced in Chemical Graph Theory by
Gutman [23].
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